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Chapter 16 Waves-I Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: August 15, 2020)
1 Introduction
1.1 Types There are two main types of waves. (i) Mechanical waves Some physical medium is being disturbed. The wave is the propagation of a disturbance through a medium. (ii) Electromagnetic waves No medium is required. Examples are light, radio waves, x-rays. All electromagnetic waves propagate in vacuum with the same speed c = 3.0 x 10^8 m/s. (iii) Matter waves (de Broglie wave in quantum mechanics). All microscopic particles such as electrons, protons, neutrons, atoms etc have a wave associated with them governed by Schrödinger’s equation.
1.2 General feature of wave In wave motion, energy is transferred over a distance. Matter is not transferred over a distance. A disturbance is transferred through space without an accompanying transfer of matter. All waves carry energy. The amount of energy and the mechanism responsible for the transport of the energy differ.
1.3 Transverse wave
A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave. The particle
motion is shown by the blue arrow, while the direction of the propagation is shown by the red arrow.
1.4 Longitudinal waves
A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave. The displacement of the coils is parallel to the propagation. The sound wave is one of examples.
1.5 Phonon in the solid (quantum mechanics of lattice vibration)
In solids, there are both longitudinal waves and transverse waves.
2 Traveling pulse The shape of the pulse at t = 0 is shown. The shape can be represented by y = f ( x ). This describes the transverse position y of the element of the string located at each value of x at t = 0. The speed of the pulse is v. At some time, t , the pulse has traveled a distance vt. The shape of the pulse does not change. The shape of the pulse at t is given by y = f ( x - vt ).
]
( ,) exp[ 2
x vt x t
with v = 1 and = 0.
(( Example-2 )) Propagation of wave packet along the x axis.
Fig. Plot of Gaussian wave packet
] 2
( ,) exp[ 2
x vt x t
with v = 1 and = 0.
as a function of x , where t is changed as a parameter, t = 0 – 1 with t = 0.25.
(( Note )) Propagation of the wave packet of the electron ( quantum mechanics ). The dispersion relation of the electron is rather different from that of light and sound. The energy is expressed by
m
k 2
ℏ^22
where ħ (= h /2) is the Dirac’s constant and h is the Planck’s constant. In this case, the probability of finding the wave packet at ( x , t ) is described by
2
2 2 4
2
2 4 2
0 2 0
2
2
]
exp[
1 ( , )
m
t k
m
t k
m
kt k x x
k
x t ℏ
1 2 3 4 5
1
2
Fig. The plot of
2
( x , t ) as a function of x , where the time t is changed as a parameter.
The position of center:
m
k t x x
The velocity of center
0 (^0) v m
k dt
d x
The width of the wave packet increases with time t.
4
2 2 1 ( ) 2
2 k m
t k
x
where = 1/ k. The amplitude of the wave packet decreases with time t ,
4
2 2 1 ( )
2 k m
t
k A
Note that y ( x , t ) is the displacement, A is the amplitude, and ( kx t ) is the phase.
Fig. Plot3D of A sin( kx - t ) with A = 1, = 1, v = 1.4.
We also have a wave function given by
y ( x , t ) A sin[ k ( x vt )]
for the travelling wave along the (- x ) direction.
Fig. Plot3D of A sin( kx + t ) with A = 1, = 1, v = 1.4.
4 The speed of a traveling wave The displacement y ( x , t ) must remain constant when the phase factor is constant.
kx t constant
We take the derivative of this equation, getting
v dt k
dx
dt
dx k
or
vk. (so- called dispersion relation)
5 Wave equation First we calculate the following from
y ( x , t ) A sin[ k ( x vt )] A sin( kx t ),
sin( )
cos( )
2 2
2 Ak kx t x
y
Ak kx t x
y
sin( )
cos( )
2 2
2 A kx t t
y
A kx t t
y
v
x t
v
x t
So that the equation for becomes
2
The solution obviously has the form
f 1 ( ) f 2 ( )
where f 1 and f 2 are arbitrary function.
or
1 ( ) 2 (^ )
v
x f t v
x
f t
The function f 1 represents a plane wave moving in the positive direction along the x axis. The function f 2 represents a plane wave moving in the negative direction along the x axis.
(( Another method )) using the Fourier transform We use the Fourier transformation technique.
x ei^ t ( x , t ) dt 2
^
x t ei^ ^^ t ( x , ) d 2
( ,) (^)
[ ( , ) ( , )] 0
2
2 2
2 2 2
2
^ ^ x d v
x x
e x v
i t
Then we have
2 (^ , )^2 ( , )^0
2
x k x x
where v
k (the dispersion relation)
The solution of this equation is
v i x ikx (^) e g e x g
^
where g ( ) is an arbitrary function of . Finally, we get
x t g e v d
i t x
( ) ( ) 2
This is an arbitrary function of ( x vt ).
6 Wave traveling in the string (transverse wave) 6.1 Simple model: the speed of waves on strings We consider a string symmetrical pulse moving from left to right along a string with speed v. We consider a reference frame, in which pulse remains stationary.
We consider one small string element of length s. The net force acting in the y direction (vertical line, toward to the origin, centripetal force) is
Fy 2 T s sin 2 Ts .
Note that s is the mass of the element and that s is equal to 2 R . At the moment, the
string element s is moving in an arc of circle. We apply the Newton’s second law to this element (centripetal force).
2 y^2 s
v s F T R
, or
2 (^2) y (^2) s v R F T R
The velocity is obtained as
v T^ s (m/s)
where tension: Ts (N=kg m/s^2 ) and (kg/m).
6.2 More general case
Fig. A snapshot of s travelling wave on a string at time t. Tension T s on the string (denoted by thick green line).
Suppose that a traveling wave is propagating along a string that is under a tension T s. Let
us consider one small element of length x. The ends of the element make small angle A
and B with the x axis. The net force acting on the element along the y -axis is
sin sin (sin sin ) (tan tan )
y s B s A s B A s B A
F T T
T T
or
2 2
[ ]
[ ]
y s B A
s s s x dx x
y y F T x x y y y y T xT xT x (^) x x x x
^
^
^ ^ ^ ^
^ ^ ^ ^
where we use the Taylor expansion. We now apply the Newton’s second law to the
element, with the mass of the element given by x x
y m x
2 1 ,
2 y y 2
y F ma x t
Then we have
2 2 2 s 2
y y x T x t x
2 2 2 2 s
y y T t x
which leads to a wave equation given by
2
2 2 2
2 2
t
y t v
y x T
y s
where
v T^ s
6.4 Energy density in wave motion Although no matter is transported down the string as the wave propagates, the energy is carried along by the wave with velocity v. As a piece of the string moves up and down executing simple harmonics, it has kinetic energy as well as potential energy (because the string is stretched like a spring).
The infinitesimal mass of the string is m = x. The kinetic energy contribution K
is given by
2 2
0
2 2
0
2 2 2 0
[ 1 cos( 2 2 )] 2
sin ( ) 2
A
A kx t dx
U dU A kx tdx
Note that U is exactly the same as K (equi-partition law of energy). The total energy in one wavelength of the wave is the sum of the kinetic energy and the potential energy,
^22
E K U 2 K A.
The power transmitted by a sinusoidal wave on a stretch string is given by
2 2 2 2 2 2 2 2 2
T A
T
Av A T
A
T
E
s
s
^ ^ .
(( Example )) Problem16- A string along which waves can travel is 2.70 m long and has a mass of 260 g. The tension in the string is 36.0 N. What must be the frequency of travelling waves of amplitude 7.70 mm for the average power to be 85.0 W?
(( Solution ))
A m
T N
kg m m
kg
s
- 70 103
^
The velocity is given by
m s
T
v s^ 19. 33 /
The average power is
Pavg v A Ts A 85 W 2
The angular frequency is
1. 24 103 rad/s,
or
f = 198 Hz.
8 Reflection and transmission of waves 8.1 Reflection of a Wave, Fixed End When the pulse reaches the support, the pulse moves back along the string in the opposite direction. This is the reflection of the pulse. The pulse is inverted when it is reflected from a fixed boundary.
8.2 Reflection of a wave, free end With a free end, the string is free to move vertically. The pulse is reflected. The pulse is not inverted when reflected from a free end.
8.3 Transmission of a wave (I)
When the waves start to overlap (b), the resultant wave function is y 1 + y 2. When crest meets crest (c) the resultant wave has a larger amplitude than either of the original waves
The two pulses separate. They continue moving in their original directions. The shapes of the pulses remain unchanged.
(( Mathematica) ) Superposition of two Gaussian wave packets traveling in the + x and – x directions
(( Example-1 ))
Fig. Plot3D of superposition of two Gaussian wave packets which is expressed by
] 2
] exp[ 2
exp[ (^2)
2 2
2
x vt x vt
, with = 0.2 and v = 1.
(( Example-2 ))
